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author | Lukaszogg <82384106+Lukaszogg@users.noreply.github.com> | 2021-07-08 20:10:11 +0200 |
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committer | Lukaszogg <82384106+Lukaszogg@users.noreply.github.com> | 2021-07-08 20:10:11 +0200 |
commit | 14033ca595b5c933caea3b214d2246529e6845b8 (patch) | |
tree | 0d6d2b2eb34e5ef5df3c517be5c1c9d803fa066c /vorlesungen/slides/7/integration.tex | |
parent | Update teil1.tex (diff) | |
parent | Only include buch.ind if it exists. (diff) | |
download | SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.tar.gz SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.zip |
Merge remote-tracking branch 'upstream/master'
Diffstat (limited to 'vorlesungen/slides/7/integration.tex')
-rw-r--r-- | vorlesungen/slides/7/integration.tex | 66 |
1 files changed, 66 insertions, 0 deletions
diff --git a/vorlesungen/slides/7/integration.tex b/vorlesungen/slides/7/integration.tex new file mode 100644 index 0000000..525e6de --- /dev/null +++ b/vorlesungen/slides/7/integration.tex @@ -0,0 +1,66 @@ +% +% integration.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Invariante Integration} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Koordinatenwechsel} +Die Koordinatentransformation +$f\colon\mathbb{R}^n\to\mathbb{R}^n:x\to y$ +hat die Ableitungsmatrix +\[ +t_{ij} += +\frac{\partial y_i}{\partial x_j} +\] +\uncover<2->{% +$n$-faches Integral +\begin{gather*} +\int\dots\int +h(f(x)) +\det +\biggl( +\frac{\partial y_i}{\partial x_j} +\biggr) +\,dx_1\,\dots dx_n +\\ += +\int\dots\int +h(y) +\,dy_1\,\dots dy_n +\end{gather*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{auf einer Lie-Gruppe} +Koordinatenwechsel sind Multiplikationen mit einer +Matrix $g\in G$ +\end{block}} +\uncover<4->{% +\begin{block}{Volumenelement in $I$} +Man muss nur das Volumenelement in $I$ in einem beliebigen +Koordinatensystem definieren: +\[ +dV = dy_1\,\dots\,dy_n +\] +\end{block}} +\uncover<5->{% +\begin{block}{Volumenelement in $g$} +\[ +\text{``\strut}g\cdot dV\text{\strut''} += +\det(g) \, dy_1\,\dots\,dy_n +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |